The Softmax function on top of a final linear layer is the de facto method to
output probability distributions in neural networks. In many applications such
as language models or text generation, this model has to produce distributions
over large output vocabularies. Recently, this has been shown to have limited
representational capacity due to its connection with the rank bottleneck in
matrix factorization. However, little is known about the limitations of
Linear-Softmax for quantities of practical interest such as cross entropy or
mode estimation, a direction that we explore here. As an efficient and
effective solution to alleviate this issue, we propose to learn parametric
monotonic functions on top of the logits. We theoretically investigate the rank
increasing capabilities of such monotonic functions. Empirically, our method
improves in two different quality metrics over the traditional Linear-Softmax
layer in synthetic and real language model experiments, adding little time or
memory overhead, while being comparable to the more computationally expensive
mixture of Softmaxes.

提出一种在 softmax 函数之上学习参数单调函数的方法，理论上和实验上都优于传统的 line-softmax 方法，在自然语言模型的应用方面具有潜在的可拓展性。

通过可学习的单调逐点非线性性消除 Softmax 瓶颈

Breaking the Softmax Bottleneck via Learnable Monotonic Pointwise  Non-linearities

Real-world machine learning applications may require functions that are
fast-to-evaluate and interpretable. In particular, guaranteed monotonicity of
the learned function can be critical to user trust. We propose meeting these
goals for low-dimensional machine learning problems by learning flexible,
monotonic functions using calibrated interpolated look-up tables. We extend the
structural risk minimization framework of lattice regression to train monotonic
look-up tables by solving a convex problem with appropriate linear inequality
constraints. In addition, we propose jointly learning interpretable
calibrations of each feature to normalize continuous features and handle
categorical or missing data, at the cost of making the objective non-convex. We
address large-scale learning through parallelization, mini-batching, and
propose random sampling of additive regularizer terms. Case studies with
real-world problems with five to sixteen features and thousands to millions of
training samples demonstrate the proposed monotonic functions can achieve
state-of-the-art accuracy on practical problems while providing greater
transparency to users.

通过学习灵活的、单调的函数以及解决凸问题的方法进行单调查找表的训练，同时积极处理连续性特征，处理分类或缺失数据，并通过并行化和小批量处理等方法实现大规模的学习，使得在实际问题上提供更高的准确性与用户可解释性。